Let $X$ be a smooth projective curve of genus $1$ in $\mathbb{P}^2(\mathbb{C})$(elliptic curve) , and $P,Q$ be two distinct points over $X$. I want to ask if there exists a curve in $\mathbb{P}^2(\mathbb{C})$ which intersects $X$ exactly at $P,Q$.
My rough idea is that the divisor $2P+2Q$ is very ample, hence by definition there exists a closed imbedding $i: X\rightarrow\mathbb{P}^n(\mathbb{C})$ such that $i^*{\mathcal{O}(1)}\cong\mathcal{O}(-2P-2Q)$, then $i^*x_k\in H^0(X,\mathcal{O}(-2P-2Q))$, which is to say $i^*x_k(\forall k=0,...,n-1)$ is a function over $X$ vanishing at $P,Q$. But there may exist other zeros of $i^*x_k$. I don't know how to continue.
Could someone give an answer?
Thanks in advance.